ECE 2105: Electromagnetic Field & Waves - Compact Class Test Study Guide

ECE 2105: CLASS TEST COMPACT REVISION GUIDE & STUDY SYSTEM

This document is a unified, self-sufficient study system created specifically for Mashuk Sir’s ECE 2105 Class Test (covering Electrostatics up to boundary conditions). It consolidates all topics from the syllabus base note, Note 01, Note 02, and Note 06.

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📋 Electrostatics Class Test Revision Checklist

1. Field Theory & Electrostatics Postulates

• Define the meaning of the word ‘field’ in terms of electromagnetics. [PYQ: 2024] [Taught in Class] (Section 1.B)
• Point out the inadequacy of circuit theory and necessity of electromagnetic field concept under high-frequency vs. quasi-static conditions. [PYQ: 2018, 2019, 2020, 2021] [Taught in Class] (Section 1.B)
• State differential/integral forms and physical significance of the two fundamental postulates of electrostatics in free space. [PYQ: 2015, 2016, 2017, 2020, 2022, 2025] [Heavily Tested] [Taught in Class] (Section 2.B)
• [DERIVATION 1] Derive the integral form of Gauss’s Law from the divergence postulate. [PYQ: 2020, 2022] [Taught in Class] (Section 2.C)
• [DERIVATION 2] Derive the integral form of KVL from the curl postulate. [PYQ: 2020, 2022] [Taught in Class] (Section 2.C)

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2. Coulomb’s Law, Gauss’s Law, and Symmetric Fields

• State Coulomb’s law and write formulas for Line, Surface, and Volume continuous charge distributions. [PYQ: 2018, 2021] [Taught in Class] (Section 3.A)
• [IMPORTANT DERIVATION 4] Derive the electric field intensity $\vec{E}$ of an infinite line charge using direct integration (Coulomb’s Law). [PYQ: 2018, 2021] [Heavily Tested] [Taught in Class] (Section 3.B)
• [DERIVATION 5] Apply Gauss’s law to determine the electric field intensity of an infinite sheet of charge. [PYQ: 2015, 2023] [Taught in Class] (Section 3.B)
• [DERIVATION 6] Apply Gauss’s law to determine the electric field intensity of an infinitely long straight line charge. [PYQ: 2016] [Taught in Class] (Section 3.B)
• [IMPORTANT DERIVATION 7] Prove that the electric field inside a uniformly charged cloud is zero at the center, varies linearly up to the surface, and decays quadratically outside. [PYQ: 2018, 2019, 2020, 2022] [Heavily Tested] [Taught in Class] (Section 3.B)

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3. Electric Potential ($V$) & Dipole

• Define electric potential ($V$), potential difference, and equipotential lines/surfaces (relation to E-field lines). [PYQ: 2016, 2017, 2021, 2024, 2025] [Taught in Class] (Section 4.A)
• [DERIVATION 8] Derive the relation $\vec{E}=-\nabla V$ from a point charge. [PYQ: 2016, 2021] [Taught in Class] (Section 4.A)
• [IMPORTANT DERIVATION 9] Derive the expression for electric potential ($V$) at a distant point due to an electric dipole. [PYQ: 2017, 2019, 2021, 2024] [Heavily Tested] [Taught in Class] (Section 4.B)
• [IMPORTANT DERIVATION 10] Derive the expression for electric field intensity ($\vec{E}$) due to an electric dipole in spherical and vector forms. [PYQ: 2016, 2023] [Taught in Class] (Section 4.B)

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4. Conductors, Dielectrics & Boundary Conditions

• State the 5 physical properties of conductors under static conditions and determine boundary fields at a free space interface. [PYQ: 2016, 2020] [Taught in Class] (Section 5.A)
• [IMPORTANT DERIVATION 11] Derive the expressions for surface polarization charge density (${\rho }_{ps}=\vec{P}\cdot {\hat{a}}_n$) and volume polarization charge density (${\rho }_p=-\nabla \cdot \vec{P}$) from first-principles potential integrals. [PYQ: 2016, 2019, 2021] [Heavily Tested] [Taught in Class] (Section 5.B)
• Show that the total electric flux density in a dielectric is $\vec{D}={ϵ}_0\vec{E}+\vec{P}$ and define constitutive permittivity relations. [PYQ: 2016, 2019, 2021] [Heavily Tested] [Taught in Class] (Section 5.C)
• [IMPORTANT DERIVATION 12] Derive boundary conditions for the tangential component of electric field $E$ at a boundary between two dielectrics ($E_{1t}=E_{2t}$). [PYQ: 2015, 2017, 2022, 2025] [Heavily Tested] [Taught in Class] (Section 6.A)
• [IMPORTANT DERIVATION 13] Derive boundary conditions for the normal component of electric flux density $D$ at a boundary between two dielectrics ($D_{1n}-D_{2n}={\rho }_s$). [PYQ: 2015, 2017, 2022, 2025] [Heavily Tested] [Taught in Class] (Section 6.B)
• [DERIVATION 14] Derive the Law of Refraction (bending) of electric field lines at a charge-free boundary. [PYQ: 2017, 2019, 2022, 2025] [Taught in Class] (Section 6.C)

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5. Poisson’s, Laplace’s & Capacitance

• [DERIVATION 15] Derive Poisson’s and Laplace’s equations for electrostatics. [PYQ: 2015, 2017, 2022, 2023, 2025] [Heavily Tested] [Taught in Class] (Section 7.A)
• [DERIVATION 16] Solve Laplace’s equation to determine the capacitance of a parallel plate capacitor ($C=\frac{ϵA}{d}$). [PYQ: 2016, 2018, 2019, 2022, 2023, 2024] [Heavily Tested] [Taught in Class] (Section 7.B)
• [IMPORTANT DERIVATION 17] Solve Laplace’s equation to determine the capacitance of a coaxial cylindrical capacitor ($C=\frac{2\pi ϵL}{\ln (b/a)}$). [PYQ: 2015, 2017] [Taught in Class] (Section 7.C)
• Estimate potential distribution and surface charge density on capacitor plates. [PYQ: 2021, 2022, 2025] [Taught in Class] (Section 7.B & 7.C)
• [DERIVATION 18] Derive the electrostatic potential energy required to assemble $k$ point charges from infinity. [PYQ: 2018, 2022, 2023, 2024, 2025] [Heavily Tested] [Taught in Class] (Section 7.D)

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6. Numericals, Traps & Sketches

• Solve the 4 classic exam numericals (dipole point charges, spherical cloud with variable density, conservative field work, and electrohydrodynamic pump). [PYQ: 2016-2025] [Taught in Class] (Section 8)
• Sketch E-field and equipotential lines for a $-{\rho }_v$ volume cloud (inward arrow trap) and an electric dipole. [PYQ: 2024, 2025] [Heavily Tested] [Taught in Class] (Section 9.A)

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1. Study Guide & Key Terms

A. Study Map: How and What to Study

• Vector Calculus Postulates (Section 2): Study divergence/curl postulates and the Cartesian proof of $\nabla \times (\nabla V)\equiv 0$. Focus on how to derive KVL from the curl postulate.

• Symmetric Gauss Applications (Section 3): Focus on the uniformly charged spherical cloud (inside vs. outside, linear growth vs. quadratic decay, maximum at surface) and the infinite line charge.

• Electric Dipole (Section 4): Practice deriving distant potential $V$ and field intensity $\vec{E}$ using binomial approximations.

• Conductors & Dielectrics (Section 5): Memorize the 5 properties of conductors. Learn the step-by-step polarization bound charge densities derivation (${\rho }_{ps}=\vec{P}\cdot {\hat{a}}_n$ and ${\rho }_p=-\nabla \cdot \vec{P}$).

• Boundary Conditions (Section 6): Derive tangential continuity $E_{1t}=E_{2t}$ and normal discontinuity $D_{1n}-D_{2n}={\rho }_s$.

• Capacitance (Section 7): Practice Parallel Plate and Cylindrical capacitor derivations using Laplace’s equation.

• Required Textbooks:

  • Field and Wave Electromagnetics (David K. Cheng) - Focuses on field theory and electromagnetic wave propagation.
  • Elements of Electromagnetics (Matthew N.O. Sadiku) - Standard reference for vector analysis and step-by-step electrostatics/magnetostatics derivations.

• Mashuk Sir Homework & Self-Study assignments:

  • Practice Exercise Problem: Ex-3.5
  • Self-Study Sections: 3.6 (Coulomb’s Law) and 3.7 (Gauss’s Law applications) from the textbook.

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B. Core Concepts Comparison

Action-at-a-Distance vs. Field Concept

• Field: {a region of space where every point is assigned a vector or scalar value representing the spatial distribution of a physical quantity} [PYQ: 2024] [Taught in Class]

• Action-at-a-Distance Concept: The outdated Newtonian model assuming that two spatially separated bodies interact instantaneously across empty space without any mediating medium or finite time delay.

• Field Concept: The modern Faraday-Maxwell model where interactions are mediated by a physical state of space (the field) which propagates through space at the finite speed of light ($c$).

• Static Electric Field: {an electric field created by stationary electric charges} The field of a static (stationary) electron ($e^{-}$). It doesn’t change with respect to time (i.e., $\frac{\partial \vec{E}}{\partial t}=0$). [Taught in Class]

• Static Magnetic Field: {a magnetic field produced by steady, time-invariant electric currents} Produced by moving electrons at a steady/DC rate (constant velocity, zero acceleration). [Taught in Class]

• Electromagnetic Waves: {self-propagating transverse oscillations of electric and magnetic fields carrying radiant energy} Produced when the movement of charges is not steady (i.e., acceleration or deceleration; time-varying currents radiate electromagnetic energy into space). [Taught in Class]

Feature	Action-at-a-Distance Concept	Field Concept (Field Theory)
Mechanism	Instantaneous force transmission through empty space without any physical intermediate medium.	Force is mediated by a physical state of space (the field) surrounding charges.
Speed of Transmission	Infinite speed (instantaneous propagation).	Finite speed (propagates at the speed of light, $c\approx 3\times 10^8\text{ m/s}$ in vacuum).
Energy Storage	Energy exists only in the massive bodies or charges themselves.	Energy and momentum are stored directly in the surrounding field space.
Applicability	Valid only for slowly moving or stationary charges/masses (static approximations).	Required for all dynamic, time-varying electromagnetic wave phenomena.

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The Inadequacy of Circuit Theory [Taught in Class]

Circuit theory (using lumped elements like $R,L,C$ and KVL/KCL) is a simplified special case of electromagnetic field theory that breaks down at high frequencies due to:

1. Finite Propagation Delay: Circuit theory assumes instantaneous signal propagation. In reality, signals travel at a finite speed $u=1/\sqrt{\mu ϵ}$. At high frequencies ($d\ge \lambda$, where $d$ is circuit size and $\lambda =u/f$), phase differences exist across components, making KVL/KCL invalid.
2. Radiation and Parasitics: High-frequency currents turn wires into antennas, causing radiation losses. Changing fields also couple traces capacitively and inductively (parasitics), causing cross-talk.

Feature	Circuit Theory	Field Theory
Primary Variables	Voltage ($V$), Current ($I$)	Electric Field ($\vec{E}$), Magnetic Field ($\vec{H}$)
Propagation Speed	Assumed infinite (instantaneous)	Finite ($u=1/\sqrt{\mu ϵ}$)
Physical Dimensions	Small compared to wavelength ($d\ll \lambda$)	Comparable to or larger than wavelength ($d\ge \lambda$)
Governing Equations	Kirchhoff’s Laws (KVL, KCL)	Maxwell’s Equations

• Quasi-Static Conditions [Taught in Class]: Operational states where the frequency is low enough ($d\ll \lambda$) that time-varying field delay is negligible ($\partial \vec{B}/\partial t\approx 0$). This allows static equations to serve as highly accurate approximations.

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2. Vector Calculus Postulates & Fundamentals

A. Curvilinear Coordinate Systems

Orthogonal curvilinear systems convert coordinates to physical distances using scale factors (metric coefficients $h_i$): $d{l}_i=h_{i}d{u}_i$.

Coordinate System	Variables ($u_1,u_2,u_3$)	Metric Coefficients ($h_1,h_2,h_3$)	Differential Volume ($dv$)
Cartesian	$(x,y,z)$	$(1,1,1)$	$dxdydz$
Cylindrical	$(r,\varphi ,z)$	$(1,r,1)$	$rdrd\varphi dz$
Spherical	$(R,\theta ,\varphi )$	$(1,R,R\sin \theta )$	$R^2\sin \theta dRd\theta d\varphi$

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B. Vector Calculus Operations: Divergence & Curl [Taught in Class]

• Divergence ($\vec{\nabla }\cdot \vec{A}$): Measures the net outward flux of a vector field per unit volume exiting an infinitesimal volume.

  • Electrostatics: $\vec{\nabla }\cdot \vec{E}=\frac{{\rho }_v}{{ϵ}_0}$ (divergence exists; charge density is source).
  • Magnetostatics: $\vec{\nabla }\cdot \vec{B}=0$ (indicating magnetic monopoles do not exist; magnetic flux loops are closed).
  • Physical Meaning:
    • $\vec{\nabla }\cdot \vec{E}>0$: Point is a source of field lines (positive charge $+Q$).
    • $\vec{\nabla }\cdot \vec{E}<0$: Point is a sink of field lines (negative charge $-Q$).

• Curl ($\vec{\nabla }\times \vec{A}$): Measures the circulation intensity or rotational vorticity of a vector field around a point.

  • Electrostatics: $\vec{\nabla }\times \vec{E}=0$ (the field is irrotational and conservative).
  • Magnetostatics: $\vec{\nabla }\times \vec{B}={\mu }_0\vec{J}$ (Ampere’s Law; magnetic fields curl around electric current density $\vec{J}$).
  • Physical Meaning: A curl of zero means work done moving a charge along any path depends only on endpoints, and closed loop work is zero (${\oint }_C\vec{E}\cdot d\vec{l}=0$).

• Divergence Theorem: {a mathematical theorem equating the volume integral of the divergence of a vector field to the closed surface integral of the flux of that field through the boundary surface} ${\int }_V(\nabla \cdot \vec{A})dv={\oint }_S\vec{A}\cdot d\vec{s}$

• Stokes’s Theorem: {a mathematical theorem equating the surface integral of the curl of a vector field to the closed line integral of that field around the boundary curve} ${\int }_S(\nabla \times \vec{A})\cdot d\vec{s}={\oint }_C\vec{A}\cdot d\vec{l}$

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C. Fundamental Postulates of Electrostatics [Taught in Class]

The behavior of static electric fields in free space is defined by two postulates:

Differential Form	Integral Form	Physical Law / Principle
$\nabla \cdot \vec{D}={\rho }_v$	${\oint }_S\vec{D}\cdot d\vec{s}=Q_{enclosed}$	Gauss’s Law (Electric flux has sources in charges)
$\nabla \times \vec{E}=0$	${\oint }_C\vec{E}\cdot d\vec{l}=0$	Conservative E-Field (Kirchhoff’s Voltage Law)

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D. High-Yield Fundamental Proofs

[DERIVATION 1] Divergence Postulate $\to$ Gauss’s Law [Taught in Class] [PYQ: 2020, 2022]

1. Start with divergence postulate: $\nabla \cdot \vec{D}={\rho }_v$
2. Integrate both sides over an arbitrary volume $V$: ${\int }_V(\nabla \cdot \vec{D})dv={\int }_V{\rho }_{v}dv$
3. Apply the Divergence Theorem (${\int }_V(\nabla \cdot \vec{A})dv={\oint }_S\vec{A}\cdot d\vec{s}$) to the left side: ${\oint }_S\vec{D}\cdot d\vec{s}={\int }_V{\rho }_{v}dv=Q_{enclosed}■$

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[DERIVATION 2] Curl Postulate $\to$ KVL (Kirchhoff’s Voltage Law) [Taught in Class] [PYQ: 2020, 2022]

1. Start with curl postulate: $\nabla \times \vec{E}=0$
2. Integrate both sides over an open surface $S$: ${\int }_S(\nabla \times \vec{E})\cdot d\vec{s}=0$
3. Apply Stokes’s Theorem (${\int }_S(\nabla \times \vec{A})\cdot d\vec{s}={\oint }_C\vec{A}\cdot d\vec{l}$) to the left side: ${\oint }_C\vec{E}\cdot d\vec{l}=0■$ Physical Meaning: The line integral of the electric field around any closed loop is zero. This corresponds directly to KVL ($\sum V=0$) in circuits.

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3. Coulomb’s Law, Gauss’s Law, and Symmetric Fields

A. Coulomb’s Law & Continuous Charge [Taught in Class]

• Coulomb’s Law: $\vec{E}={\hat{a}}_R\frac{q}{4\pi {ϵ}_0{R}^2}$
  • Definition: {states that the electrostatic force (or mediating electric field intensity $\vec{E}$) between static point charges in free space is directly proportional to the product of charges and inversely proportional to the square of distance between them} [PYQ: 2018, 2021] [Taught in Class]
• Continuous Charge distributions: Used when charge is distributed over a region of space rather than localizing on point charges:
  • Line Charge: $\vec{E}=\frac{1}{4\pi {ϵ}_0}{\int }_{L^{\prime }}\frac{{\rho }_{L}d{l}^{\prime }}{R^2}{\hat{a}}_R$ where ${\rho }_L$ is line charge density $\{\text{C/m}\}$.
  • Surface Charge: $\vec{E}=\frac{1}{4\pi {ϵ}_0}{\int }_{S^{\prime }}\frac{{\rho }_{S}d{s}^{\prime }}{R^2}{\hat{a}}_R$ where ${\rho }_S$ is surface charge density $\{{\text{C/m}}^2\}$.
  • Volume Charge: $\vec{E}=\frac{1}{4\pi {ϵ}_0}{\int }_{V^{\prime }}\frac{{\rho }_{v}d{v}^{\prime }}{R^2}{\hat{a}}_R$ where ${\rho }_v$ is volume charge density $\{{\text{C/m}}^3\}$.

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B. Gauss’s Law & Symmetric Field Proofs

• Gauss’s Law: {the total outward electric flux through any closed surface is equal to the total free charge enclosed by that surface divided by the permittivity of the medium: ${\oint }_S\vec{E}\cdot d\vec{s}=\frac{Q_{enclosed}}{{ϵ}_0}$} [PYQ: 2015, 2016, 2019, 2020, 2023] [Taught in Class]
• Gaussian Surface: {an imaginary closed 3D surface matching the symmetry of a charge distribution, such that the electric field magnitude $E$ is constant and directed parallel or perpendicular to the surface normal vector at every point} [Taught in Class]

[IMPORTANT DERIVATION 4] Infinite Line Charge Field via Direct Integration (Coulomb’s Law) [Taught in Class] [PYQ: 2018, 2021]

Problem: Derive the electric field intensity $\vec{E}$ at distance $r$ from an infinitely long straight line charge of density ${\rho }_L$ on the $z$-axis.

1. Coordinate Setup (Write in Exam):

   • Line charge lies along the $z$-axis. Observation point is $P(r,0,0)$.
   • A charge element $d{q}^{\prime }={\rho }_{L}d{z}^{\prime }$ is located at source coordinate $(0,0,z^{\prime })$.
   • Vectors: ${\vec{R}}^{\prime }={\hat{a}}_z{z}^{\prime }$, $\vec{R}={\hat{a}}_{r}r$, $\vec{R}-{\vec{R}}^{\prime }={\hat{a}}_{r}r-{\hat{a}}_z{z}^{\prime }$.
   • Distance magnitude: $R=\sqrt{r^2+z^{\prime 2}}$, Unit vector: ${\hat{a}}_R=\frac{{\hat{a}}_{r}r-{\hat{a}}_z{z}^{\prime }}{\sqrt{r^2+z^{\prime 2}}}$.

2. Set up the Integral (Write in Exam): $d\vec{E}=\frac{d{q}^{\prime }}{4\pi {ϵ}_0{R}^2}{\hat{a}}_R=\frac{{\rho }_L({\hat{a}}_{r}r-{\hat{a}}_z{z}^{\prime })d{z}^{\prime }}{4\pi {ϵ}_0(r^2+z^{\prime 2}{)}^{3/2}}$ Integrate from $z^{\prime }=-\infty$ to $+\infty$: $\vec{E}=\frac{{\rho }_L}{4\pi {ϵ}_0}{\int }_{-\infty }^{\infty }\frac{{\hat{a}}_{r}r-{\hat{a}}_z{z}^{\prime }}{(r^2+z^{\prime 2}{)}^{3/2}}d{z}^{\prime }$

3. Symmetry Cancellation (Write in Exam): Due to symmetry along the infinite $z$-axis, longitudinal $z$-components cancel out ($E_z=0$). This leaves only the radial field $E_r$: $E_r=\frac{{\rho }_{L}r}{4\pi {ϵ}_0}{\int }_{-\infty }^{\infty }\frac{d{z}^{\prime }}{(r^2+z^{\prime 2}{)}^{3/2}}$

   Study Detail: Proof of Symmetry Cancellation $f(z^{\prime })=\frac{z^{\prime }}{(r^2+z^{\prime 2}{)}^{3/2}}$ is an odd function. Integrating an odd function over symmetric limits $(-\infty ,\infty )$ yields exactly zero: $E_z=-\frac{{\rho }_L}{4\pi {ϵ}_0}{\int }_{-\infty }^{\infty }\frac{z^{\prime }d{z}^{\prime }}{(r^2+z^{\prime 2}{)}^{3/2}}=0■$

   Integrand

4. Solve the Integral (Write in Exam): Evaluate the radial integral using standard formulas: $E_r=\frac{{\rho }_{L}r}{4\pi {ϵ}_0}{\left[\frac{z^{\prime }}{r^2\sqrt{r^2+z^{\prime 2}}}\right]}_{-\infty }^{\infty }=\frac{{\rho }_{L}r}{4\pi {ϵ}_0}\left(\frac{2}{r^2}\right)=\frac{{\rho }_L}{2\pi {ϵ}_{0}r}$ In vector form: $\vec{\mathbf{E}}={\hat{\mathbf{a}}}_{\mathbf{r}}\frac{{\rho }_{\mathbf{L}}}{2\pi {ϵ}_0\mathbf{r}}(\text{V/m})■$

   [!tip] Study Detail: Trig Substitution Solving Steps

   Substitute $z^{\prime }=r\tan \theta ,d{z}^{\prime }=r{\sec }^2\theta d\theta$, limits $\theta \to \pm \pi /2$:

   ${\int }_{-\infty }^{\infty }\frac{d{z}^{\prime }}{(r^2+z^{\prime 2}{)}^{3/2}}={\int }_{-\pi /2}^{\pi /2}\frac{r{\sec }^2\theta d\theta }{r^3{\sec }^3\theta }=\frac{1}{r^2}{\int }_{-\pi /2}^{\pi /2}\cos \theta d\theta =\frac{1}{r^2}[\sin \theta {]}_{-\pi /2}^{\pi /2}=\frac{2}{r^2}■$

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[DERIVATION 5] Infinite Sheet of Charge Field via Gauss’s Law [Taught in Class] [PYQ: 2015, 2023]

Problem: Derive the electric field of an infinite flat plane with uniform surface charge density ${\rho }_s$ in the $z=0$ plane.

1. Gaussian Surface Setup (Write in Exam):
   • Choose a circular cylinder Gaussian surface of area $A$ extending symmetrically normal to the sheet from $z=-h$ to $z=+h$.
   • The field must point only normal to the sheet ($E_z{\hat{a}}_z$ for $z>0$, and $-E_z{\hat{a}}_z$ for $z<0$).
2. Evaluate Gauss Flux (Write in Exam): ${\oint }_S\vec{D}\cdot d\vec{s}={\int }_{top}\vec{D}\cdot d\vec{s}+{\int }_{bottom}\vec{D}\cdot d\vec{s}=D_{z}A+D_{z}A=2D_{z}A$ (No flux crosses the curved side surface because $D$ is parallel to it).
3. Equate to Enclosed Charge (Write in Exam): $2D_{z}A=Q_{enclosed}={\rho }_{s}A⟹D_z=\frac{{\rho }_s}{2}$ In terms of electric field intensity $\vec{E}$: $\vec{\mathbf{E}}={\hat{\mathbf{a}}}_{\mathbf{n}}\frac{{\rho }_{\mathbf{s}}}{2{ϵ}_0}(\text{V/m})■$

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[DERIVATION 6] Infinitely Long Line Charge Field via Gauss’s Law [Taught in Class] [PYQ: 2016]

Problem: Derive the electric field of an infinite line charge of density ${\rho }_L$ on the $z$-axis using Gauss’s Law.

1. Gaussian Surface Setup (Write in Exam):
   • Choose a coaxial cylinder of radius $r$ and length $L$.
   • The field is purely radial: $\vec{E}=E_r{\hat{a}}_r$. No flux crosses the flat end caps.
2. Evaluate Gauss Flux & Enclosed Charge (Write in Exam): ${\oint }_S\vec{D}\cdot d\vec{s}=D_r(2\pi rL)$ $Q_{enclosed}={\rho }_{L}L$
3. Equate & Solve (Write in Exam): $D_r(2\pi rL)={\rho }_{L}L⟹D_r=\frac{{\rho }_L}{2\pi r}⟹\vec{\mathbf{E}}={\hat{\mathbf{a}}}_{\mathbf{r}}\frac{{\rho }_{\mathbf{L}}}{2\pi {ϵ}_0\mathbf{r}}■$

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[IMPORTANT DERIVATION 7] Uniformly Charged Spherical Cloud [Taught in Class] [PYQ: 2018, 2019, 2020, 2022]

Problem: Prove the field of a spherical cloud of radius $a$ with uniform volume charge density ${\rho }_0$ is zero at its center, grows linearly inside, and decays quadratically outside.

1. Case 1: Inside the Cloud ($R\le a$) (Write in Exam):
   • Choose a concentric Gaussian sphere of radius $R<a$.
   • Flux: ${\oint }_S\vec{D}\cdot d\vec{s}=D_R(4\pi R^2)$
   • Enclosed Charge: $Q_{enclosed}={\int }_0^R{\rho }_{0}dv={\rho }_0\left(\frac{4}{3}\pi R^3\right)$
   • Equate both sides: $D_R(4\pi R^2)={\rho }_0\left(\frac{4}{3}\pi R^3\right)⟹D_R=\frac{{\rho }_{0}R}{3}⟹{\vec{\mathbf{E}}}_{\mathbf{in}}={\hat{\mathbf{a}}}_{\mathbf{R}}\frac{{\rho }_0\mathbf{R}}{3{ϵ}_0}■$
2. Case 2: Outside the Cloud ($R>a$) (Write in Exam):
   • Choose a concentric Gaussian sphere of radius $R>a$.
   • Flux: ${\oint }_S\vec{D}\cdot d\vec{s}=D_R(4\pi R^2)$
   • Enclosed Charge: The entire cloud charge is enclosed: $Q_{enclosed}={\rho }_0\left(\frac{4}{3}\pi a^3\right)$
   • Equate both sides: $D_R(4\pi R^2)={\rho }_0\left(\frac{4}{3}\pi a^3\right)⟹D_R=\frac{{\rho }_0{a}^3}{3R^2}⟹{\vec{\mathbf{E}}}_{\mathbf{out}}={\hat{\mathbf{a}}}_{\mathbf{R}}\frac{{\rho }_0{\mathbf{a}}^3}{3{ϵ}_0{\mathbf{R}}^2}■$
3. Evaluate at Surface boundary ($R=a$) (Write in Exam): $E_{in}(a)=\frac{{\rho }_{0}a}{3{ϵ}_0}\text{and}{E}_{out}(a)=\frac{{\rho }_0{a}^3}{3{ϵ}_0{a}^2}=\frac{{\rho }_{0}a}{3{ϵ}_0}$ Since the field increases linearly for $R\le a$ and decreases quadratically for $R>a$, the field is zero at the center ($R=0$), and reaches its absolute maximum at the surface ($R=a$): $E_{max}=\frac{{\rho }_{0}a}{3{ϵ}_0}$

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4. Electric Potential ($V$) & Dipole

A. Definitions & Field-Potential Relations

• Electric Field Intensity ($\vec{E}$): {the force exerted per unit positive charge placed at a point in space: $\vec{E}={\lim }_{q\to 0}\frac{\vec{F}}{q}$, representing force mediating fields} [PYQ: 2016, 2019, 2021] [Taught in Class]

• Electric Potential ($V$): {the scalar potential field representing electric potential energy per unit charge, equal to the work required to carry a unit positive charge from infinity to a point in space} [PYQ: 2016, 2017, 2021] [Taught in Class] $V=-{\int }_{\infty }^P\vec{E}\cdot d\vec{l}$

• Electric Potential Difference: {the work required to move a unit positive charge between two points $P_1$ and $P_2$ in an electric field} [PYQ: 2017] [Taught in Class] $V_2-V_1=-{\int }_{P_1}^{P_2}\vec{E}\cdot d\vec{l}$

• Gradient Relation: $\vec{E}=-\nabla V$ (The negative sign signifies that the electric field $\vec{E}$ points in the direction of maximum rate of decrease of the potential, i.e., from high voltage to low voltage).

  [DERIVATION 8] Electric Field from Scalar Potential ($\vec{E}=-\nabla V$) [Taught in Class] [PYQ: 2016, 2021]

  1. Potential of a point charge: $V=\frac{q}{4\pi {ϵ}_{0}R}$
  2. Take gradient in spherical coordinates: $\nabla V={\hat{a}}_R\frac{\partial V}{\partial R}={\hat{a}}_R\frac{\partial }{\partial R}\left(\frac{q}{4\pi {ϵ}_{0}R}\right)=-{\hat{a}}_R\frac{q}{4\pi {ϵ}_0{R}^2}=-\vec{E}⟹\vec{E}=-\nabla V■$

• Equipotential Lines [Taught in Class]: Contours of equal potential ($dV=0$). Moving a charge along an equipotential contour requires zero work ($dW=-qdV=0$). Equipotential lines and electric field lines are always strictly perpendicular at every point.

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B. High-Yield Dipole Proofs

• Electric Dipole: {consists of two equal and opposite point charges $+q$ and $-q$ separated by a very small distance $d$} [PYQ: 2019, 2024] [Taught in Class]
• Electric Dipole Moment ($\vec{p}$): {a vector representing the strength and orientation of the dipole, defined as $\vec{p}=q\vec{d}$, pointing from negative charge $-q$ to positive charge $+q$} [PYQ: 2016, 2021, 2023] [Taught in Class]

[IMPORTANT DERIVATION 9] Potential of an Electric Dipole [Taught in Class] [PYQ: 2017, 2019, 2021, 2024]

Problem: Derive the potential $V$ at a distant point $P(R,\theta )$ from two equal and opposite charges $+q$ and $-q$ separated by distance $d$.

1. Potential Formulation (Write in Exam): $V=\frac{q}{4\pi {ϵ}_0}\left(\frac{1}{R_{+}}-\frac{1}{R_{-}}\right)$

2. Taylor/Binomial Approximation (Write in Exam):

   • For $R\gg d$, we write the approximate inverse distances: $\frac{1}{R_{+}}\approx \frac{1}{R}\left(1+\frac{d}{2R}\cos \theta \right)\text{and}\frac{1}{R_{-}}\approx \frac{1}{R}\left(1-\frac{d}{2R}\cos \theta \right)$

   Study Detail: Geometric Law of Cosines Expansion Steps $R_{\pm }^2=R^2+(d/2{)}^2\mp Rd\cos \theta \approx R^2\left(1\mp \frac{d}{R}\cos \theta \right)$ Take the inverse root: $\frac{1}{R_{\pm }}=R^{-1}{\left(1\mp \frac{d}{R}\cos \theta \right)}^{-1/2}\approx \frac{1}{R}\left(1\pm \frac{d}{2R}\cos \theta \right)■$

3. Subtract and Solve (Write in Exam): $\frac{1}{R_{+}}-\frac{1}{R_{-}}\approx \frac{1}{R}\left[\left(1+\frac{d}{2R}\cos \theta \right)-\left(1-\frac{d}{2R}\cos \theta \right)\right]=\frac{d\cos \theta }{R^2}$ $V=\frac{qd\cos \theta }{4\pi {ϵ}_0{R}^2}=\frac{\vec{\mathbf{p}}\cdot {\hat{\mathbf{a}}}_{\mathbf{R}}}{4\pi {ϵ}_0{\mathbf{R}}^2}\text{(V)}■$ where $\vec{p}=q\vec{d}={\hat{a}}_{z}qd$ is the electric dipole moment vector.

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[IMPORTANT DERIVATION 10] Field Intensity of an Electric Dipole [Taught in Class] [PYQ: 2016, 2023]

• Method 1: Gradient in Spherical Coordinates (Write in Exam): $\vec{E}=-\nabla V=-\left[{\hat{a}}_R\frac{\partial V}{\partial R}+{\hat{a}}_{\theta }\frac{1}{R}\frac{\partial V}{\partial \theta }\right]$ $E_R=-\frac{\partial }{\partial R}\left(\frac{p\cos \theta }{4\pi {ϵ}_0{R}^2}\right)=\frac{2p\cos \theta }{4\pi {ϵ}_0{R}^3}$ $E_{\theta }=-\frac{1}{R}\frac{\partial }{\partial \theta }\left(\frac{p\cos \theta }{4\pi {ϵ}_0{R}^2}\right)=\frac{p\sin \theta }{4\pi {ϵ}_0{R}^3}$ $\vec{\mathbf{E}}=\frac{\mathbf{p}}{4\pi {ϵ}_0{\mathbf{R}}^3}\left(2\cos \theta {\hat{\mathbf{a}}}_{\mathbf{R}}+\sin \theta {\hat{\mathbf{a}}}_{\theta }\right)(\text{V/m})■$

• Method 2: Vector Form Approximation (Write in Exam): For $R\gg d$, we approximate the denominators: $|\vec{R}\pm \vec{d}/2{|}^{-3}\approx R^{-3}\left(1\mp \frac{3}{2}\frac{\vec{R}\cdot \vec{d}}{R^2}\right)$

  Study Detail: Denominator Taylor Steps $|\vec{R}\pm \vec{d}/2{|}^{-3}={\left(R^2\pm \vec{R}\cdot \vec{d}\right)}^{-3/2}\approx R^{-3}{\left(1\pm \frac{\vec{R}\cdot \vec{d}}{R^2}\right)}^{-3/2}\approx R^{-3}\left(1\mp \frac{3}{2}\frac{\vec{R}\cdot \vec{d}}{R^2}\right)■$

  Substitute back into the Coulomb vector form: $\vec{E}\approx \frac{q}{4\pi {ϵ}_0{R}^3}\left[\left(\vec{R}-\frac{\vec{d}}{2}\right)\left(1+\frac{3}{2}\frac{\vec{R}\cdot \vec{d}}{R^2}\right)-\left(\vec{R}+\frac{\vec{d}}{2}\right)\left(1-\frac{3}{2}\frac{\vec{R}\cdot \vec{d}}{R^2}\right)\right]$ Simplifying and neglecting $d^2$ terms yields: $\vec{\mathbf{E}}=\frac{1}{4\pi {ϵ}_0{\mathbf{R}}^3}\left[\frac{3(\vec{\mathbf{p}}\cdot \vec{\mathbf{R}})}{{\mathbf{R}}^2}\vec{\mathbf{R}}-\vec{\mathbf{p}}\right]■$

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5. Conductors & Dielectrics in Static Fields

A. Conductors in Static Fields [Taught in Class]

• Conductor: {a material possessing an abundance of free mobile electrons, allowing electric charge to flow easily under the influence of an applied field} [Taught in Class]

Under static conditions, an ideal conductor has:

1. Zero Internal Field: ${\vec{E}}_{in}=0$ (any internal electric field exerts forces on free electrons, moving them to the surface until they create an opposing field that cancels the internal field).
2. Zero Internal Charge: ${\rho }_v=0$ (by Gauss’s Law, all net charges reside on the surface).
3. Equipotential Volume: $V_a=V_b$ inside and on the surface (Interior of a static conductor is equipotential).
4. Perpendicular Boundary Field: Tangential component $E_t=0$; normal component $E_n={\rho }_s/{ϵ}_0$.

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B. Dielectric Polarization & Bound Charges

• Dielectric: {an insulating material containing no free mobile charges, but whose bound charges (electrons and nuclei) displace slightly under an external electric field to form microscopic dipoles} [Taught in Class]

• Polarization Vector ($\vec{P}$): {the net electric dipole moment per unit volume of a dielectric material: $\vec{P}={\lim }_{\Delta V\to 0}\frac{\sum {\vec{p}}_k}{\Delta V}$, measured in ${\text{C/m}}^2$} [PYQ: 2016, 2019, 2021] [Taught in Class] External fields align molecular dipoles, creating polarization.

  [IMPORTANT DERIVATION 11] Dielectric Polarization Potential & Bound Charge Densities [Taught in Class] [PYQ: 2016, 2019, 2021]

  Problem: Derive surface bound charge ${\rho }_{ps}$ and volume bound charge ${\rho }_p$.

  1. Set up potential integral (Write in Exam): $V=\frac{1}{4\pi {ϵ}_0}{\int }_{V^{\prime }}\frac{\vec{P}\cdot {\hat{a}}_R}{R^2}d{v}^{\prime }$ 2. Substitute gradient identity (Write in Exam): Since ${\hat{a}}_R/R^2={\nabla }^{\prime }(1/R)$: $V=\frac{1}{4\pi {ϵ}_0}{\int }_{V^{\prime }}\vec{P}\cdot {\nabla }^{\prime }\left(\frac{1}{R}\right)d{v}^{\prime }$ 3. Expand via vector product rule (Write in Exam): $V=\frac{1}{4\pi {ϵ}_0}\left[{\int }_{V^{\prime }}{\nabla }^{\prime }\cdot \left(\frac{\vec{P}}{R}\right)d{v}^{\prime }-{\int }_{V^{\prime }}\frac{{\nabla }^{\prime }\cdot \vec{P}}{R}d{v}^{\prime }\right]$ 4. Apply Divergence Theorem to the first term (Write in Exam): $V=\frac{1}{4\pi {ϵ}_0}{\oint }_{S^{\prime }}\frac{\vec{P}\cdot {\hat{a}}_n}{R}d{S}^{\prime }+\frac{1}{4\pi {ϵ}_0}{\int }_{V^{\prime }}\frac{-({\nabla }^{\prime }\cdot \vec{P})}{R}d{v}^{\prime }$ 5. Match Charge Densities (Write in Exam): Comparing to standard potentials, we define:

  • Surface Bound Charge Density (${\rho }_{ps}$): {bound charge density appearing on the dielectric surface: ${\rho }_{ps}=\vec{P}\cdot {\hat{a}}_n$}
  • Volume Bound Charge Density (${\rho }_p$): {bound charge density accumulated inside the dielectric volume: ${\rho }_p=-\nabla \cdot \vec{P}$}

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C. Electric Flux Density ($\vec{D}$) & Medium Relations [Taught in Class]

• Electric Flux Density ($\vec{D}$): {also called displacement field; a vector quantity representing the density of electric flux from free charges, independent of medium polarization: $\vec{D}={ϵ}_0\vec{E}+\vec{P}$, measured in ${\text{C/m}}^2$} [PYQ: 2022, 2025] [Taught in Class]
• Constitutive Relations: For linear and isotropic dielectrics: $\vec{P}={ϵ}_0{\chi }_e\vec{E}$ $\vec{D}={ϵ}_0(1+{\chi }_e)\vec{E}={ϵ}_0{ϵ}_r\vec{E}=ϵ\vec{E}$ where:
  • ${\chi }_e$ is the dimensionless Electric Susceptibility {measuring dielectric polarizability}.
  • ${ϵ}_r=1+{\chi }_e$ is the Relative Permittivity {or dielectric constant}.
  • $ϵ={ϵ}_0{ϵ}_r$ is the Permittivity of the medium.

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6. Electrostatic Boundary Conditions

Crossing an interface between two media changes the fields according to boundary rules:

A. Tangential Component ($E_{1t}=E_{2t}$)

[IMPORTANT DERIVATION 12] Tangential Electric Field Boundary Condition [Taught in Class] [PYQ: 2015, 2017, 2022, 2025]

1. Loop Integration (Write in Exam): Apply irrotational postulate ${\oint }_C\vec{E}\cdot d\vec{l}=0$ over a rectangular loop $abcd$ of length $\Delta w$ and height $\Delta h$ crossing the boundary.
2. Let height $\Delta h\to 0$: $\oint \vec{E}\cdot d\vec{l}=E_{1t}\Delta w-E_{2t}\Delta w=0⟹{\mathbf{E}}_{1t}={\mathbf{E}}_{2t}■$ Tangential component of electric field is continuous across any interface.

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B. Normal Component ($D_{1n}-D_{2n}={\rho }_s$)

[IMPORTANT DERIVATION 13] Normal Electric Flux Density Boundary Condition [Taught in Class] [PYQ: 2015, 2017, 2022, 2025]

1. Pillbox Integration (Write in Exam): Apply Gauss’s Law ${\oint }_S\vec{D}\cdot d\vec{s}=Q_{enclosed}$ to a small cylinder of area $\Delta s$ and height $\Delta h$ crossing the boundary.
2. Let height $\Delta h\to 0$: $\oint \vec{D}\cdot d\vec{s}=D_{1n}\Delta s-D_{2n}\Delta s={\rho }_s\Delta s⟹{\mathbf{D}}_{1n}-{\mathbf{D}}_{2n}={\rho }_{\mathbf{s}}■$ (If boundary has no free charge, ${\rho }_s=0⟹D_{1n}=D_{2n}$).

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C. Refraction of Field Lines

[DERIVATION 14] Bending (Refraction) of Electric Field Lines [Taught in Class] [PYQ: 2017, 2019, 2022, 2025]

At a charge-free interface (${\rho }_s=0$): $E_{1t}=E_{2t}⟹E_1\sin {\alpha }_1=E_2\sin {\alpha }_2$ $D_{1n}=D_{2n}⟹{ϵ}_1{E}_1\cos {\alpha }_1={ϵ}_2{E}_2\cos {\alpha }_2$ Dividing tangential by normal yields the Law of Refraction: $\frac{\tan {\alpha }_1}{\tan {\alpha }_2}=\frac{{ϵ}_1}{{ϵ}_2}■$

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7. Poisson’s, Laplace’s, and Capacitance

A. Poisson’s & Laplace’s Equations

[DERIVATION 15] Poisson’s and Laplace’s Equations [Taught in Class] [PYQ: 2015, 2017, 2022, 2023, 2025]

1. Gauss’s Law: $\nabla \cdot \vec{D}={\rho }_v⟹\nabla \cdot (ϵ\vec{E})={\rho }_v$
2. For homogeneous medium: $\nabla \cdot \vec{E}=\frac{{\rho }_v}{ϵ}$
3. Substitute $\vec{E}=-\nabla V$: $\nabla \cdot (-\nabla V)=\frac{{\rho }_v}{ϵ}⟹{\nabla }^2\mathbf{V}=-\frac{{\rho }_{\mathbf{v}}}{ϵ}\text{(Poisson’s Equation)}■$
4. In charge-free regions (${\rho }_v=0$): ${\nabla }^2\mathbf{V}=0\text{(Laplace’s Equation)}■$

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B. Parallel Plate Capacitor Capacitance

[DERIVATION 16] Capacitance of a Parallel Plate Capacitor [Taught in Class] [PYQ: 2016, 2018, 2019, 2022, 2023, 2024]

Problem: Derive the capacitance of parallel plates of area $A$ separated by distance $d$ at potentials $V(0)=0$ and $V(d)=V_0$.

1. Solve Laplace’s Equation (Write in Exam): $\frac{d^{2}V}{d{y}^2}=0⟹V(y)=C_{1}y+C_2$
2. Apply Boundary Conditions (Write in Exam):
   • $V(0)=0⟹C_2=0$.
   • $V(d)=V_0⟹C_1=V_0/d$.
   • Potential: $V(y)=\frac{V_0}{d}y$.
3. Find Charge and Capacitance (Write in Exam):
   • Field: $\vec{E}=-\nabla V=-{\hat{a}}_y\frac{V_0}{d}$.
   • Surface charge: ${\rho }_s=D_n=ϵE_n=\frac{ϵV_0}{d}$.
   • Capacitance: $C=\frac{Q}{V_0}=\frac{{\rho }_{s}A}{V_0}⟹\mathbf{C}=\frac{ϵ\mathbf{A}}{\mathbf{d}}(\text{F})■$

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C. Cylindrical Capacitor Capacitance

[IMPORTANT DERIVATION 17] Capacitance of a Coaxial Cylindrical Capacitor [Taught in Class] [PYQ: 2015, 2017]

Problem: Derive the capacitance of coaxial cylinders of radii $a$ and $b$ ($a<b$), length $L$, at potentials $V(a)=V_0$ and $V(b)=0$.

1. Solve Laplace’s Equation (Write in Exam): $\frac{1}{r}\frac{d}{dr}\left(r\frac{dV}{dr}\right)=0⟹V(r)=C_1\ln (r)+C_2$
2. Apply Boundary Conditions (Write in Exam):
   • $V(b)=0⟹C_2=-C_1\ln (b)$.
   • $V(a)=V_0⟹C_1=-\frac{V_0}{\ln (b/a)}$.
   • Potential distribution: $V(r)=V_0\frac{\ln (b/r)}{\ln (b/a)}$.
3. Find Charge and Capacitance (Write in Exam):
   • Field: $\vec{E}=-{\hat{a}}_r\frac{dV}{dr}={\hat{a}}_r\frac{V_0}{r\ln (b/a)}$.
   • Surface charge at inner conductor ($r=a$): ${\rho }_s=ϵE_n(a)=\frac{ϵV_0}{a\ln (b/a)}$.
   • Total charge: $Q={\rho }_s(2\pi aL)=\frac{2\pi ϵL{V}_0}{\ln (b/a)}$.
   • Capacitance: $C=\frac{Q}{V_0}⟹\mathbf{C}=\frac{2\pi ϵ\mathbf{L}}{\ln (b/a)}(\text{F})■$

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D. Electrostatic Energy (Assembling Point Charges)

• Electrostatic Potential Energy ($W_e$): {the potential energy stored in a static configuration of electric charges, equal to the work required to assemble those charges one by one from infinity} [PYQ: 2018, 2019, 2020, 2022, 2023, 2024, 2025] [Taught in Class]

[DERIVATION 18] Electrostatic Energy Stored in Point Charge Assembly [Taught in Class] [PYQ: 2018, 2022, 2023, 2024, 2025]

The work required to assemble $k$ point charges one by one from infinity is stored as potential energy: $W_e=\frac{1}{2}{\sum }_{i=1}^k{q}_i{V}_i\text{where }{V}_i=\frac{1}{4\pi {ϵ}_0}{\sum }_{j\ne i}\frac{q_j}{R_{ij}}$

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8. Step-by-Step Solutions to Classic Exam Numericals

Numerical 1: Dipole Axis Point Charges [Taught in Class]

Question: A charge $q_1=-2\mu \text{C}$ is at the origin $(0,0,0)$. Two positive charges $q_2=q_3=+1\mu \text{C}$ are at $y=\pm 2\text{ m}$. Calculate potential $V$ and field strength $\vec{E}$ at point $P(4,0,0)$.

1. Calculate potential $V$ (Scalar sum):
   • Distances: $R_1=4\text{ m}$, $R_2=R_3=\sqrt{4^2+2^2}=\sqrt{20}=2\sqrt{5}\text{ m}$.
   • Potential: $V=\frac{1}{4\pi {ϵ}_0}\left(\frac{q_1}{R_1}+\frac{q_2}{R_2}+\frac{q_3}{R_3}\right)$
   • $V=9\times 10^9\left(\frac{-2\times 10^{-6}}{4}+\frac{1\times 10^{-6}}{2\sqrt{5}}+\frac{1\times 10^{-6}}{2\sqrt{5}}\right)=9\times 10^3\left(-0.5+\frac{1}{\sqrt{5}}\right)\approx -475.2\text{ V}$.
2. Calculate field strength $\vec{E}$ (Vector sum):
   • By symmetry, $y$-directed components cancel: $E_y=0$.
   • Radial component from $q_1$: $E_{1x}=-\frac{9\times 10^9\times 2\times 10^{-6}}{4^2}=-1125\text{ V/m}$.
   • Radial component from $q_2,q_3$: $\cos \theta =4/\sqrt{20}$. $E_{2x}=E_{3x}=\frac{9\times 10^9\times 1\times 10^{-6}}{20}\times \frac{4}{\sqrt{20}}\approx 40.25\text{ V/m}$
   • Total field: $\vec{E}={\hat{a}}_x(-1125+40.25+40.25)=-{\hat{a}}_{x}1044.5\text{ V/m}$.

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Numerical 2: Variable Density Spherical Charge Cloud [Taught in Class]

Question: A spherical charge cloud in free space has ${\rho }_v=2r{\text{ nC/m}}^3$ for $0\le r\le 10\text{ m}$ and zero otherwise. Calculate $\vec{E}$ at $r=2\text{ m}$ and $r=12\text{ m}$.

1. Field at $r=2\text{ m}$ (Inside, $r<10\text{ m}$):
   • Apply Gauss: $D_r(4\pi r^2)={\int }_0^r(2r^{\prime })(4\pi r^{\prime 2}d{r}^{\prime })=2\pi r^4$
   • $D_r=\frac{2\pi r^4}{4\pi r^2}=\frac{r^2}{2}{\text{ nC/m}}^2$. At $r=2\text{ m}$: $D_r=2{\text{ nC/m}}^2$.
   • $\vec{E}={\hat{a}}_r\frac{2\times 10^{-9}}{8.854\times 10^{-12}}={\hat{a}}_{r}225.9\text{ V/m}$.
2. Field at $r=12\text{ m}$ (Outside, $r>10\text{ m}$):
   • Total charge: $Q_{total}={\int }_0^{10}(2r^{\prime })(4\pi r^{\prime 2}d{r}^{\prime })=2\pi [r^{\prime 4}{]}_0^{10}=20000\pi \text{ nC}$.
   • $D_r=\frac{Q_{total}}{4\pi r^2}=\frac{20000\pi \times 10^{-9}}{4\pi (12{)}^2}=34.72{\text{ nC/m}}^2$.
   • $\vec{E}={\hat{a}}_r\frac{34.72\times 10^{-9}}{8.854\times 10^{-12}}={\hat{a}}_{r}3921.4\text{ V/m}$.

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Numerical 3: Work Done in an Inhomogeneous Field [Taught in Class]

Question: Determine work done in carrying a $-2\mu \text{C}$ charge from $P_1(2,1,-1)$ to $P_2(8,2,-1)$ in field $\vec{E}={\hat{a}}_{x}y+{\hat{a}}_{y}x\text{ V/m}$.

1. Line integral setup: $\vec{E}\cdot d\vec{l}=ydx+xdy=d(xy)\text{(Perfect differential! Field is conservative)}$
2. Integrate over endpoints: ${\int }_{P_1}^{P_2}\vec{E}\cdot d\vec{l}=[xy{]}_{(2,1)}^{(8,2)}=(8\times 2)-(2\times 1)=16-2=14\text{ V}$
3. Work done by external force: $W=-q{\int }_{P_1}^{P_2}\vec{E}\cdot d\vec{l}=-(-2\times 10^{-6}\text{ C})\times 14\text{ V}=2.8\times 10^{-5}\text{ J}=28\mu \text{J}■$

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Numerical 4: Electrohydrodynamic Pump Potential Derivation [Taught in Class]

Question: Electrode region has uniform charge density ${\rho }_0$. Left electrode at $x=0$ has potential $V_0$, right electrode at $x=d$ has potential $0\text{ V}$. Find expressions for $V$ and $\vec{E}$ at any point.

1. Solve Poisson’s Equation: $\frac{d^{2}V}{d{x}^2}=-\frac{{\rho }_0}{ϵ}⟹V(x)=-\frac{{\rho }_0}{2ϵ}{x}^2+C_{1}x+C_2$
2. Apply Boundary Conditions:
   • $V(0)=V_0⟹C_2=V_0$.
   • $V(d)=0⟹-\frac{{\rho }_0}{2ϵ}{d}^2+C_{1}d+V_0=0⟹C_1=\frac{{\rho }_{0}d}{2ϵ}-\frac{V_0}{d}$.
3. Write potential and field equations: $V(x)=V_0\left(1-\frac{x}{d}\right)+\frac{{\rho }_0}{2ϵ}x(d-x)■$ $\vec{E}(x)=-\frac{dV}{dx}{\hat{a}}_x={\hat{a}}_x\left[\frac{V_0}{d}+\frac{{\rho }_0}{ϵ}\left(x-\frac{d}{2}\right)\right]■$

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9. Cheatsheet & Pitfalls

A. Visual Sketches & Traps

• The Inward Negative Sphere Trap [Taught in Class]: Drawing arrows outward for a $-{\rho }_v$ charge sphere. Because the charge density is negative, the field points radially inward toward the center.
• Equipotentials & Field lines orthogonality [Taught in Class]: Always make sure your equipotential contours (concentric dashed circles or ovals) intersect the solid electric field lines at exactly a $90^{\circ }$ perpendicular angle.

                 NEGATIVE SPHERICAL CLOUD SKETCH (-ρ_v)
                 
                           E-Field Lines: INWARD (solid)
                           Equipotentials: concentric circles (dashed)
                                     
                                     \   |   /
                                      v  |  v
                                   .-----|-----.
                                  /   \  |  /   \
                                 /     v | v     \
                                |------> (O) <----|
                                 \     ^ | ^     /
                                  \   /  |  \   /
                                   '-----|-----'
                                      ^  |  ^
                                     /   |   \

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B. Core Exam Hacks

1. Work Done External Force Sign: Always use the formula $W=-q\int \vec{E}\cdot d\vec{l}$. If the charge is negative and the integral is positive, the work is positive (requires physical input from the external agent to move against field forces).
2. Gaussian Surface Justification: When starting a Gauss’s Law derivation, always state: “We select a Gaussian surface $S$ matching the symmetry of the charge distribution, such that $\vec{D}$ is everywhere either parallel or perpendicular to the normal vector $d\vec{s}$.” This ensures full presentation marks.
3. Units check: Never omit units in final solutions:
   • Electric Field ($\vec{E}$): $\text{V/m}$
   • Electric Potential ($V$): $\text{V}$ (Volts)
   • Capacitance ($C$): $\text{F}$ (Farads)
   • Charge densities: ${\rho }_L$ ($\text{C/m}$), ${\rho }_s$ (${\text{C/m}}^2$), ${\rho }_v$ (${\text{C/m}}^3$).